5

Points]DETAILSThe area of the surface obtained by rotating the curveV 64-x2 -8 < x < 8 about the X-axis isSubmnit AnswerView Previous QuestionHome...

Question

Points]DETAILSThe area of the surface obtained by rotating the curveV 64-x2 -8 < x < 8 about the X-axis isSubmnit AnswerView Previous QuestionHome

Points] DETAILS The area of the surface obtained by rotating the curve V 64-x2 -8 < x < 8 about the X-axis is Submnit Answer View Previous Question Home



Answers

Use a CAS to find the exact area of the surface generated by revolving the curve about the stated axis. $$ 8 x y^{2}=2 y^{6}+1,1 \leq y \leq 2 ; y \text { -axis } $$

Which number four X is equal to immigration from it will be by it for pets one plus if dash off ext. It's quit yet she's equal toe from 1 to 2. Boy export three figuration Want lost nine x but for the X, where ever packs equal to X over three. If this were to be ext over to Hey, Rick with one and B is equal to two. So this is equal to the creation from 1 to 2 or by over 36. Immigration off you. You where u is equal to one plus nine Next hour. Four to be you went to 36 x over three The D X, which is like with for over 18. Immigration from one group you won't half you So it's immigration is able to boy over 18 You 3/2 but people were true. 12 after this institution and return back that you will be put one plus nine x over work. So when negative fire over 27 crews were load off 10 minus 29 squared off once were five boy

So here we're going to find the area of the surface generated by rotating this curve about the X axis Between the bounds of x equals one, x equals three. And to do that we're gonna be using our area of a the surface area of a revolution solid or revolved curve however you want to say it. And so first thing we're gonna need is to get our derivative in terms of X. So we're going to use some quotient rule here. This is gonna be derivative over the top, times the bottom minus diverted at the bottom minus the directive at the bottom minus the top, all over the bottom squared. So let's do some simplifying. We got 48 Next to the seven minus 16, Next to the seven minus 32 X. All over 64 x to the fore. They fixed this for it's bugging me. So let's go ahead and take out a 32 from top and the bottom. After we subtract these two, We get next to the 7 -1 Over two X. of the four. So here's what we're gonna use for our area integral. So don't quite have enough space. I just go ahead and start down here. So we're gonna take the integral from 1 to 3 times our why? Or are ffx next to the six plus 2/8 X. Squared. And now we're gonna plug in our Square root of one plus the derivative squared Next to 7 -1 Over two X. of the four. And it's going to be dx. All right. So the first thing I'm gonna do is expand this inside. Ah Let's do the whole thing once. I'm gonna expand this entire inside the radical and see what I can get. So let's put a star here and the star here. So, if we expand the X seven minus X over two X. To the forest squared, We'll get next to the 14 -2 x. plus X squared. All over four actually eight. So this is going to be and then we have a plus one. So I'm gonna go ahead and change this into for extra eight Over four weeks of the 8 to get a common denominator. And then this will add with our middle term and then in the numerator, So these two ad And so -2-plus 4 will give us a positive too. So this becomes Next to the 14 Plus two extra 8 plus X squared All over four times next to it. And so this numerator will now factor in the same way it did before in our original ah area integral. But this time it will have a positive second term. So finally we can write this ass Next to the seven plus x. All over to exit before all squared. And so this is what is inside our square root. And if you'd like you can foil this out to sea, you'll get the same result. So let's continue with our integral. And you go from 1 to 3 Next the 6-plus 2 Over eight x squared times. Now this is what we have here. I'm going to go ahead and cancel this radical. Going to cancel this radical with our square. So what we're left with is simply what's inside the square here Next to the seven plus x. Over two x. 2 four D. X. Now let's multiply this out. Next to the six times extra seven. Give me an extra 13 Plus two extra 7 Plus X to the seven plus two X. This is all over 16 extra six. The X. Go ahead and combine these middle terms without rewriting the entire thing. This will be three X 27. And so we can just Distribute the 16 X to the six between all the terms. Oh I forgot my Sorry about that. I dropped my two pi along the way. We put this back in. Remember in our formula we have a two pi out front. So now we're gonna get a bunch of Broken up terms. So we have to pay items integral from 1-3. This is going to be extra seven over 16 plus three x over 16 plus 1/8 X. The five. This is all dX. So let's go ahead and evaluate are integral. So integral of X to 7/16. What's going to be add one the power and then divide by the new power. So it's gonna be over one, Right, 80 plus 48. Yeah And then integral of three x. over 16. It's going to be three X squared over 32. And now this is actually 1/8 X. to the -5. So we add one power and then divide by the new power, Add 1 to the power, divide by the new power. So we have 1/8. Next the -4 over negative four. And you can rewrite this as 1/32 times x to the fore the negative sign out front just like that. I forgot my evaluation bar From 1- three. So let's do some plugging in. Ah We will have 3 to the eight which I cannot evaluate in my head. Plus three squared is nine times 3 is 27 Minour through the 4th which is 81. 32 times 81. Let's just say there is that put in some more brackets, this is minus. We have ones everywhere, which makes life a lot easier. Plus 3/32 minus 1/32. And so you can evaluate this all out And don't forget to multiply by two pi at the end. You should get, we have an exact answer. You will get exactly 8429 over 81. Hi, there you are.

We know that we're gonna be using the formula the integral to pie from 0 to 2 times. Why, Diaz, which, in other words, essentially means we're gonna be having X cubed times square of one plus three x squared squared, which is nine X to the fourth D backs. Okay, now we know we can do some substitution. Over here you is one plus nine extra fourth, which means X cubed de acts is do you divide by 36? Which means we have two pi times integral from 1 to 1 45 you to the 1/2. Do you okay? Use the power method to integrate, which means increased the experiment by Juan divide by the new exponents. Now you're going to be plugging in our upper and are lower bounds. The lower bound plugged in. Just 2/3. You can see this is the upper ground plug. Done. This simplifies to pi over 27 times 1 45 times squared of 1 45 minus one

This question asked us to find exact area of the surface by rotating the curb about the X axis. As it stated, What we know we need to do is we know we need to first off, figure out the bounds. They've actually given this to us of exes between zero and three. This means zero and three are about. Now we know we pull out to pie because we know we're gonna be rotating it as it's stated in the problem. And then we know that we have Why squared is X plus one. What we know this means is that if axes y squared minus one, then D axe over D. Y is equivalent to why this is critical here. Because now what we know we have is remember when we're writing this were essentially plugging in using X. So that is the same thing. That's why you can consider this to be. And then we know that we need to integrate. So in order to do that, we know we're going to be We know we're going to be using the for the multiplication pattern. Essentially distribution eight times people see, is a B plus a C So it's from 0 to 3 to pie X plus five over four de axe. We know we're going to be integrating this. We use the power rule of increasing the exploded by one dividing by the new exponents. That's how you integrate. And then we plug in and what we end up with is pi. Over six time 17 squared of 17 minus five squared of five.


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